When I started my study of Latin I already knew that the articles had to be supplied but as I gain experience translating texts it seems just about any word can be omitted if the context is clear. Did the Romans speak this way in casual conversation?

You can see the same thing in modern languages that are highly inflected, as Latin was. In English, much of the meaning comes from word order - a word by itself cannot convey its grammatical role in the sentence; instead, it needs other words to convey the grammar. In Latin, a single word can convey a wealth of grammatical meaning, implying its grammatical context more clearly than a stand-alone word can do in English. Highly inflected modern languages, like the Slavic languages (Russian, Polish, etc.) for example, work the same way, allowing people to leave out words in casual conversation without any resulting ambiguity.

As I progress in my studies of Latin I can't help but compare it to German which I already know. For instance German is still a high inflected language but at the same time observes fairly strict word order and when conjugating verbs the pronouns are always used; for instance a person would always say Ich gehe(I am going) never just gehe. It would seem to me that this defeats the purpose of an inflected language.